19

I'd avoid giving problems like that to students first learning indefinite integrals (either by not asking it at all, or specifying the range x>1 in the question). It's a subtle algebraic trap, and if the goal is to teach students the mechanics of integration, it's going to be distracting rather than helpful. It might be an interesting question in a more ...


11

You are right to be concerned that the students are "missing something", but IMO the real problem here is that the question is completely artificial. In any application of this type of integral, most likely $x$ will be known to be either positive or negative, but not both, and only one part of the "either-or" answer would apply. And there had better be a ...


11

This is a hard question, because students are so used to manipulation of this kind. I have found you are right that absolute values can cause the worst of these examples. Here is an example I ran into recently, which I hope will help your thinking. Observe that there are two different limits here: $$\lim_{x\to\pm\infty} \frac{x}{\sqrt{x^2+1}} = \pm 1$$ ...


10

One of the charms of graph theory is that people of all ages often enjoy learning graph theory ideas and tools. One place one can read about these ideas is in the book called For All Practical Purposes (I am a co-author) which has gone through 10 editions. The book was designed for college liberal arts students who might have almost no proficiency with ...


7

To me, the key point here is that the integral runs over a singularity. If you naively calculates a definite form that runs over the singularity you get the wrong answer. This is something I have done enough so that I have taught myself to be careful in this case. I am more a physicist than a mathematician, so what I care about is the connection to a ...


4

Here are GeoGebra materials: Graph Theory for Kids, inspired by Joel Hamkins' notes, to which @A.Goodier pointed.                     Four-color challenge.


3

This sort of example is not artificial and needs more careful treatment than it often receives. The inherent difficulty of the example is compounded by the tendency to discuss primitives without discussing their domains. The badly named "indefinite integral" of a function is really an incompletely defined primitive of a function. By "incompletely defined" I ...


2

It is interesting to note that if we instead write the antiderivative as $I=-\arctan\left(\frac{1}{\sqrt{x^2-1}}\right)$, then this form is valid for both $x<1$ and $x>1$. In other words, using this arctan representation, we avoid a need for a piecewise representation for the antiderivative. This problem is well suited for formative assessment. I ...


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